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sinh^2(x)-5cosh(x)=-7

Solution

sinh^{2} (x) - 5 cosh (x) = - 7

Solution

x = ln (0.17157 \dots), x = ln (0.26794 \dots), x = ln (3.73205 \dots), x = ln (5.82842 \dots)

Degrees

x = - 100.99797 \dots^{\circ}, x = - 75.45612 \dots^{\circ}, x = 75.45612 \dots^{\circ}, x = 100.99797 \dots^{\circ}

Solution steps

sinh^{2} (x) - 5 cosh (x) = - 7

Rewrite using trig identities

sinh^{2} (x) - 5 cosh (x) = - 7

Use the Hyperbolic identity:

sinh (x) = \frac{e ^{x} - e ^{- x}}{2}

(\frac{e ^{x} - e ^{- x}}{2})^{2} - 5 cosh (x) = - 7

Use the Hyperbolic identity:

cosh (x) = \frac{e ^{x} + e ^{- x}}{2}

(\frac{e ^{x} - e ^{- x}}{2})^{2} - 5 \cdot \frac{e ^{x} + e ^{- x}}{2} = - 7

(\frac{e ^{x} - e ^{- x}}{2})^{2} - 5 \cdot \frac{e ^{x} + e ^{- x}}{2} = - 7

(\frac{e ^{x} - e ^{- x}}{2})^{2} - 5 \cdot \frac{e ^{x} + e ^{- x}}{2} = - 7 : x = ln (0.17157 \dots), x = ln (0.26794 \dots), x = ln (3.73205 \dots), x = ln (5.82842 \dots)

(\frac{e ^{x} - e ^{- x}}{2})^{2} - 5 \cdot \frac{e ^{x} + e ^{- x}}{2} = - 7

Apply exponent rules

(\frac{e ^{x} - e ^{- x}}{2})^{2} - 5 \cdot \frac{e ^{x} + e ^{- x}}{2} = - 7

Apply exponent rule:

a^{b c} = (a^{b})^{c}

e^{- x} = (e^{x})^{- 1}

(\frac{e ^{x} - ( e ^{x} ) ^{- 1}}{2})^{2} - 5 \cdot \frac{e ^{x} + ( e ^{x} ) ^{- 1}}{2} = - 7

(\frac{e ^{x} - ( e ^{x} ) ^{- 1}}{2})^{2} - 5 \cdot \frac{e ^{x} + ( e ^{x} ) ^{- 1}}{2} = - 7

Rewrite the equation with

e^{x} = u

(\frac{u - ( u ) ^{- 1}}{2})^{2} - 5 \cdot \frac{u + ( u ) ^{- 1}}{2} = - 7

Solve

(\frac{u - u ^{- 1}}{2})^{2} - 5 \cdot \frac{u + u ^{- 1}}{2} = - 7 : u \approx 0.17157 \dots, u \approx 0.26794 \dots, u \approx 3.73205 \dots, u \approx 5.82842 \dots

(\frac{u - u ^{- 1}}{2})^{2} - 5 \cdot \frac{u + u ^{- 1}}{2} = - 7

Refine

\frac{( u ^{2} - 1 ) ^{2}}{4 u ^{2}} - \frac{5 ( u ^{2} + 1 )}{2 u} = - 7

Multiply by LCM

\frac{( u ^{2} - 1 ) ^{2}}{4 u ^{2}} - \frac{5 ( u ^{2} + 1 )}{2 u} = - 7

Find Least Common Multiplier of

4 u^{2}, 2 u : 4 u^{2}

4 u^{2}, 2 u

Lowest Common Multiplier (LCM)

Least Common Multiplier of

4, 2 : 4

4, 2

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Multiply each factor the greatest number of times it occurs in either

4

2

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

Compute an expression comprised of factors that appear either in

4 u^{2}

2 u

= 4 u^{2}

Multiply by LCM=

4 u^{2}

\frac{( u ^{2} - 1 ) ^{2}}{4 u ^{2}} \cdot 4 u^{2} - \frac{5 ( u ^{2} + 1 )}{2 u} \cdot 4 u^{2} = - 7 \cdot 4 u^{2}

Simplify

\frac{( u ^{2} - 1 ) ^{2}}{4 u ^{2}} \cdot 4 u^{2} - \frac{5 ( u ^{2} + 1 )}{2 u} \cdot 4 u^{2} = - 7 \cdot 4 u^{2}

Simplify

\frac{( u ^{2} - 1 ) ^{2}}{4 u ^{2}} \cdot 4 u^{2} : (u^{2} - 1)^{2}

\frac{( u ^{2} - 1 ) ^{2}}{4 u ^{2}} \cdot 4 u^{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( u ^{2} - 1 ) ^{2} \cdot 4 u ^{2}}{4 u ^{2}}

Cancel the common factor:

4

= \frac{( u ^{2} - 1 ) ^{2} u ^{2}}{u ^{2}}

Cancel the common factor:

u^{2}

= (u^{2} - 1)^{2}

Simplify

- \frac{5 ( u ^{2} + 1 )}{2 u} \cdot 4 u^{2} : - 10 u (u^{2} + 1)

- \frac{5 ( u ^{2} + 1 )}{2 u} \cdot 4 u^{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= - \frac{5 ( u ^{2} + 1 ) \cdot 4 u ^{2}}{2 u}

Multiply the numbers:

5 \cdot 4 = 20

= - \frac{20 u ^{2} ( u ^{2} + 1 )}{2 u}

Divide the numbers:

\frac{20}{2} = 10

= \frac{10 u ^{2} ( u ^{2} + 1 )}{u}

Cancel the common factor:

u

= - 10 u (u^{2} + 1)

Simplify

- 7 \cdot 4 u^{2} : - 28 u^{2}

- 7 \cdot 4 u^{2}

Multiply the numbers:

7 \cdot 4 = 28

= - 28 u^{2}

(u^{2} - 1)^{2} - 10 u (u^{2} + 1) = - 28 u^{2}

(u^{2} - 1)^{2} - 10 u (u^{2} + 1) = - 28 u^{2}

(u^{2} - 1)^{2} - 10 u (u^{2} + 1) = - 28 u^{2}

Solve

(u^{2} - 1)^{2} - 10 u (u^{2} + 1) = - 28 u^{2} : u \approx 0.17157 \dots, u \approx 0.26794 \dots, u \approx 3.73205 \dots, u \approx 5.82842 \dots

(u^{2} - 1)^{2} - 10 u (u^{2} + 1) = - 28 u^{2}

Expand

(u^{2} - 1)^{2} - 10 u (u^{2} + 1) : u^{4} - 2 u^{2} + 1 - 10 u^{3} - 10 u

(u^{2} - 1)^{2} - 10 u (u^{2} + 1)

(u^{2} - 1)^{2} : u^{4} - 2 u^{2} + 1

Apply Perfect Square Formula:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = u^{2}, b = 1

= (u^{2})^{2} - 2 u^{2} \cdot 1 + 1^{2}

Simplify

(u^{2})^{2} - 2 u^{2} \cdot 1 + 1^{2} : u^{4} - 2 u^{2} + 1

(u^{2})^{2} - 2 u^{2} \cdot 1 + 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= (u^{2})^{2} - 2 \cdot 1 \cdot u^{2} + 1

(u^{2})^{2} = u^{4}

(u^{2})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= u^{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= u^{4}

2 u^{2} \cdot 1 = 2 u^{2}

2 u^{2} \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 2 u^{2}

= u^{4} - 2 u^{2} + 1

= u^{4} - 2 u^{2} + 1

= u^{4} - 2 u^{2} + 1 - 10 u (u^{2} + 1)

Expand

- 10 u (u^{2} + 1) : - 10 u^{3} - 10 u

- 10 u (u^{2} + 1)

Apply the distributive law:

a (b + c) = ab + a c

a = - 10 u, b = u^{2}, c = 1

= - 10 u u^{2} + (- 10 u) \cdot 1

Apply minus-plus rules

+ (- a) = - a

= - 10 u^{2} u - 10 \cdot 1 \cdot u

Simplify

- 10 u^{2} u - 10 \cdot 1 \cdot u : - 10 u^{3} - 10 u

- 10 u^{2} u - 10 \cdot 1 \cdot u

10 u^{2} u = 10 u^{3}

10 u^{2} u

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

u^{2} u = u^{2 + 1}

= 10 u^{2 + 1}

Add the numbers:

2 + 1 = 3

= 10 u^{3}

10 \cdot 1 \cdot u = 10 u

10 \cdot 1 \cdot u

Multiply the numbers:

10 \cdot 1 = 10

= 10 u

= - 10 u^{3} - 10 u

= - 10 u^{3} - 10 u

= u^{4} - 2 u^{2} + 1 - 10 u^{3} - 10 u

u^{4} - 2 u^{2} + 1 - 10 u^{3} - 10 u = - 28 u^{2}

Move

28 u^{2}

to the left side

u^{4} - 2 u^{2} + 1 - 10 u^{3} - 10 u = - 28 u^{2}

Add

28 u^{2}

to both sides

u^{4} - 2 u^{2} + 1 - 10 u^{3} - 10 u + 28 u^{2} = - 28 u^{2} + 28 u^{2}

Simplify

u^{4} - 10 u^{3} + 26 u^{2} - 10 u + 1 = 0

u^{4} - 10 u^{3} + 26 u^{2} - 10 u + 1 = 0

Find one solution for

u^{4} - 10 u^{3} + 26 u^{2} - 10 u + 1 = 0

using Newton-Raphson:

u \approx 0.17157 \dots

u^{4} - 10 u^{3} + 26 u^{2} - 10 u + 1 = 0

Newton-Raphson Approximation Definition

f (u) = u^{4} - 10 u^{3} + 26 u^{2} - 10 u + 1

Find

f^{^{'}} (u) : 4 u^{3} - 30 u^{2} + 52 u - 10

\frac{d}{d u} (u^{4} - 10 u^{3} + 26 u^{2} - 10 u + 1)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d u} (u^{4}) - \frac{d}{d u} (10 u^{3}) + \frac{d}{d u} (26 u^{2}) - \frac{d}{d u} (10 u) + \frac{d}{d u} (1)

\frac{d}{d u} (u^{4}) = 4 u^{3}

\frac{d}{d u} (u^{4})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 4 u^{4 - 1}

Simplify

= 4 u^{3}

\frac{d}{d u} (10 u^{3}) = 30 u^{2}

\frac{d}{d u} (10 u^{3})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 10 \frac{d}{d u} (u^{3})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 10 \cdot 3 u^{3 - 1}

Simplify

= 30 u^{2}

\frac{d}{d u} (26 u^{2}) = 52 u

\frac{d}{d u} (26 u^{2})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 26 \frac{d}{d u} (u^{2})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 26 \cdot 2 u^{2 - 1}

Simplify

= 52 u

\frac{d}{d u} (10 u) = 10

\frac{d}{d u} (10 u)

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 10 \frac{d u}{d u}

Apply the common derivative:

\frac{d u}{d u} = 1

= 10 \cdot 1

Simplify

= 10

\frac{d}{d u} (1) = 0

\frac{d}{d u} (1)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= 4 u^{3} - 30 u^{2} + 52 u - 10 + 0

Simplify

= 4 u^{3} - 30 u^{2} + 52 u - 10

Let

u_{0} = 0

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 0.1 : Δ u_{1} = 0.1

f (u_{0}) = 0^{4} - 10 \cdot 0^{3} + 26 \cdot 0^{2} - 10 \cdot 0 + 1 = 1

f^{^{'}} (u_{0}) = 4 \cdot 0^{3} - 30 \cdot 0^{2} + 52 \cdot 0 - 10 = - 10

u_{1} = 0.1

Δ u_{1} = ∣ 0.1 - 0 ∣ = 0.1

Δ u_{1} = 0.1

u_{2} = 0.14907 \dots : Δ u_{2} = 0.04907 \dots

f (u_{1}) = 0. 1^{4} - 10 \cdot 0. 1^{3} + 26 \cdot 0. 1^{2} - 10 \cdot 0.1 + 1 = 0.2501

f^{^{'}} (u_{1}) = 4 \cdot 0. 1^{3} - 30 \cdot 0. 1^{2} + 52 \cdot 0.1 - 10 = - 5.096

u_{2} = 0.14907 \dots

Δ u_{2} = ∣ 0.14907 \dots - 0.1 ∣ = 0.04907 \dots

Δ u_{2} = 0.04907 \dots

u_{3} = 0.16783 \dots : Δ u_{3} = 0.01875 \dots

f (u_{2}) = 0.14907 \dots^{4} - 10 \cdot 0.14907 \dots^{3} + 26 \cdot 0.14907 \dots^{2} - 10 \cdot 0.14907 \dots + 1 = 0.05441 \dots

f^{^{'}} (u_{2}) = 4 \cdot 0.14907 \dots^{3} - 30 \cdot 0.14907 \dots^{2} + 52 \cdot 0.14907 \dots - 10 = - 2.90143 \dots

u_{3} = 0.16783 \dots

Δ u_{3} = ∣ 0.16783 \dots - 0.14907 \dots ∣ = 0.01875 \dots

Δ u_{3} = 0.01875 \dots

u_{4} = 0.17143 \dots : Δ u_{4} = 0.00360 \dots

f (u_{3}) = 0.16783 \dots^{4} - 10 \cdot 0.16783 \dots^{3} + 26 \cdot 0.16783 \dots^{2} - 10 \cdot 0.16783 \dots + 1 = 0.00755 \dots

f^{^{'}} (u_{3}) = 4 \cdot 0.16783 \dots^{3} - 30 \cdot 0.16783 \dots^{2} + 52 \cdot 0.16783 \dots - 10 = - 2.09886 \dots

u_{4} = 0.17143 \dots

Δ u_{4} = ∣ 0.17143 \dots - 0.16783 \dots ∣ = 0.00360 \dots

Δ u_{4} = 0.00360 \dots

u_{5} = 0.17157 \dots : Δ u_{5} = 0.00014 \dots

f (u_{4}) = 0.17143 \dots^{4} - 10 \cdot 0.17143 \dots^{3} + 26 \cdot 0.17143 \dots^{2} - 10 \cdot 0.17143 \dots + 1 = 0.00027 \dots

f^{^{'}} (u_{4}) = 4 \cdot 0.17143 \dots^{3} - 30 \cdot 0.17143 \dots^{2} + 52 \cdot 0.17143 \dots - 10 = - 1.94704 \dots

u_{5} = 0.17157 \dots

Δ u_{5} = ∣ 0.17157 \dots - 0.17143 \dots ∣ = 0.00014 \dots

Δ u_{5} = 0.00014 \dots

u_{6} = 0.17157 \dots : Δ u_{6} = 2.13816 E - 7

f (u_{5}) = 0.17157 \dots^{4} - 10 \cdot 0.17157 \dots^{3} + 26 \cdot 0.17157 \dots^{2} - 10 \cdot 0.17157 \dots + 1 = 4.15046 E - 7

f^{^{'}} (u_{5}) = 4 \cdot 0.17157 \dots^{3} - 30 \cdot 0.17157 \dots^{2} + 52 \cdot 0.17157 \dots - 10 = - 1.94113 \dots

u_{6} = 0.17157 \dots

Δ u_{6} = ∣ 0.17157 \dots - 0.17157 \dots ∣ = 2.13816 E - 7

Δ u_{6} = 2.13816 E - 7

u \approx 0.17157 \dots

Apply long division:

\frac{u ^{4} - 10 u ^{3} + 26 u ^{2} - 10 u + 1}{u - 0.17157 \dots} = u^{3} - 9.82842 \dots u^{2} + 24.31370 \dots u - 5.82842 \dots

u^{3} - 9.82842 \dots u^{2} + 24.31370 \dots u - 5.82842 \dots \approx 0

Find one solution for

u^{3} - 9.82842 \dots u^{2} + 24.31370 \dots u - 5.82842 \dots = 0

using Newton-Raphson:

u \approx 0.26794 \dots

u^{3} - 9.82842 \dots u^{2} + 24.31370 \dots u - 5.82842 \dots = 0

Newton-Raphson Approximation Definition

f (u) = u^{3} - 9.82842 \dots u^{2} + 24.31370 \dots u - 5.82842 \dots

Find

f^{^{'}} (u) : 3 u^{2} - 19.65685 \dots u + 24.31370 \dots

\frac{d}{d u} (u^{3} - 9.82842 \dots u^{2} + 24.31370 \dots u - 5.82842 \dots)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d u} (u^{3}) - \frac{d}{d u} (9.82842 \dots u^{2}) + \frac{d}{d u} (24.31370 \dots u) - \frac{d}{d u} (5.82842 \dots)

\frac{d}{d u} (u^{3}) = 3 u^{2}

\frac{d}{d u} (u^{3})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 3 u^{3 - 1}

Simplify

= 3 u^{2}

\frac{d}{d u} (9.82842 \dots u^{2}) = 19.65685 \dots u

\frac{d}{d u} (9.82842 \dots u^{2})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 9.82842 \dots \frac{d}{d u} (u^{2})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 9.82842 \dots \cdot 2 u^{2 - 1}

Simplify

= 19.65685 \dots u

\frac{d}{d u} (24.31370 \dots u) = 24.31370 \dots

\frac{d}{d u} (24.31370 \dots u)

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 24.31370 \dots \frac{d u}{d u}

Apply the common derivative:

\frac{d u}{d u} = 1

= 24.31370 \dots \cdot 1

Simplify

= 24.31370 \dots

\frac{d}{d u} (5.82842 \dots) = 0

\frac{d}{d u} (5.82842 \dots)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= 3 u^{2} - 19.65685 \dots u + 24.31370 \dots - 0

Simplify

= 3 u^{2} - 19.65685 \dots u + 24.31370 \dots

Let

u_{0} = 0

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 0.23971 \dots : Δ u_{1} = 0.23971 \dots

f (u_{0}) = 0^{3} - 9.82842 \dots \cdot 0^{2} + 24.31370 \dots \cdot 0 - 5.82842 \dots = - 5.82842 \dots

f^{^{'}} (u_{0}) = 3 \cdot 0^{2} - 19.65685 \dots \cdot 0 + 24.31370 \dots = 24.31370 \dots

u_{1} = 0.23971 \dots

Δ u_{1} = ∣ 0.23971 \dots - 0 ∣ = 0.23971 \dots

Δ u_{1} = 0.23971 \dots

u_{2} = 0.26758 \dots : Δ u_{2} = 0.02786 \dots

f (u_{1}) = 0.23971 \dots^{3} - 9.82842 \dots \cdot 0.23971 \dots^{2} + 24.31370 \dots \cdot 0.23971 \dots - 5.82842 \dots = - 0.55101 \dots

f^{^{'}} (u_{1}) = 3 \cdot 0.23971 \dots^{2} - 19.65685 \dots \cdot 0.23971 \dots + 24.31370 \dots = 19.77400 \dots

u_{2} = 0.26758 \dots

Δ u_{2} = ∣ 0.26758 \dots - 0.23971 \dots ∣ = 0.02786 \dots

Δ u_{2} = 0.02786 \dots

u_{3} = 0.26794 \dots : Δ u_{3} = 0.00036 \dots

f (u_{2}) = 0.26758 \dots^{3} - 9.82842 \dots \cdot 0.26758 \dots^{2} + 24.31370 \dots \cdot 0.26758 \dots - 5.82842 \dots = - 0.00705 \dots

f^{^{'}} (u_{2}) = 3 \cdot 0.26758 \dots^{2} - 19.65685 \dots \cdot 0.26758 \dots + 24.31370 \dots = 19.26866 \dots

u_{3} = 0.26794 \dots

Δ u_{3} = ∣ 0.26794 \dots - 0.26758 \dots ∣ = 0.00036 \dots

Δ u_{3} = 0.00036 \dots

u_{4} = 0.26794 \dots : Δ u_{4} = 6.27517 E - 8

f (u_{3}) = 0.26794 \dots^{3} - 9.82842 \dots \cdot 0.26794 \dots^{2} + 24.31370 \dots \cdot 0.26794 \dots - 5.82842 \dots = - 1.20873 E - 6

f^{^{'}} (u_{3}) = 3 \cdot 0.26794 \dots^{2} - 19.65685 \dots \cdot 0.26794 \dots + 24.31370 \dots = 19.26206 \dots

u_{4} = 0.26794 \dots

Δ u_{4} = ∣ 0.26794 \dots - 0.26794 \dots ∣ = 6.27517 E - 8

Δ u_{4} = 6.27517 E - 8

u \approx 0.26794 \dots

Apply long division:

\frac{u ^{3} - 9.82842 \dots u ^{2} + 24.31370 \dots u - 5.82842 \dots}{u - 0.26794 \dots} = u^{2} - 9.56047 \dots u + 21.75198 \dots

u^{2} - 9.56047 \dots u + 21.75198 \dots \approx 0

Find one solution for

u^{2} - 9.56047 \dots u + 21.75198 \dots = 0

using Newton-Raphson:

u \approx 3.73205 \dots

u^{2} - 9.56047 \dots u + 21.75198 \dots = 0

Newton-Raphson Approximation Definition

f (u) = u^{2} - 9.56047 \dots u + 21.75198 \dots

Find

f^{^{'}} (u) : 2 u - 9.56047 \dots

\frac{d}{d u} (u^{2} - 9.56047 \dots u + 21.75198 \dots)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d u} (u^{2}) - \frac{d}{d u} (9.56047 \dots u) + \frac{d}{d u} (21.75198 \dots)

\frac{d}{d u} (u^{2}) = 2 u

\frac{d}{d u} (u^{2})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 u^{2 - 1}

Simplify

= 2 u

\frac{d}{d u} (9.56047 \dots u) = 9.56047 \dots

\frac{d}{d u} (9.56047 \dots u)

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 9.56047 \dots \frac{d u}{d u}

Apply the common derivative:

\frac{d u}{d u} = 1

= 9.56047 \dots \cdot 1

Simplify

= 9.56047 \dots

\frac{d}{d u} (21.75198 \dots) = 0

\frac{d}{d u} (21.75198 \dots)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= 2 u - 9.56047 \dots + 0

Simplify

= 2 u - 9.56047 \dots

Let

u_{0} = 2

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 3.19252 \dots : Δ u_{1} = 1.19252 \dots

f (u_{0}) = 2^{2} - 9.56047 \dots \cdot 2 + 21.75198 \dots = 6.63103 \dots

f^{^{'}} (u_{0}) = 2 \cdot 2 - 9.56047 \dots = - 5.56047 \dots

u_{1} = 3.19252 \dots

Δ u_{1} = ∣ 3.19252 \dots - 2 ∣ = 1.19252 \dots

Δ u_{1} = 1.19252 \dots

u_{2} = 3.64038 \dots : Δ u_{2} = 0.44785 \dots

f (u_{1}) = 3.19252 \dots^{2} - 9.56047 \dots \cdot 3.19252 \dots + 21.75198 \dots = 1.42212 \dots

f^{^{'}} (u_{1}) = 2 \cdot 3.19252 \dots - 9.56047 \dots = - 3.17542 \dots

u_{2} = 3.64038 \dots

Δ u_{2} = ∣ 3.64038 \dots - 3.19252 \dots ∣ = 0.44785 \dots

Δ u_{2} = 0.44785 \dots

u_{3} = 3.72836 \dots : Δ u_{3} = 0.08798 \dots

f (u_{2}) = 3.64038 \dots^{2} - 9.56047 \dots \cdot 3.64038 \dots + 21.75198 \dots = 0.20057 \dots

f^{^{'}} (u_{2}) = 2 \cdot 3.64038 \dots - 9.56047 \dots = - 2.27971 \dots

u_{3} = 3.72836 \dots

Δ u_{3} = ∣ 3.72836 \dots - 3.64038 \dots ∣ = 0.08798 \dots

Δ u_{3} = 0.08798 \dots

u_{4} = 3.73204 \dots : Δ u_{4} = 0.00367 \dots

f (u_{3}) = 3.72836 \dots^{2} - 9.56047 \dots \cdot 3.72836 \dots + 21.75198 \dots = 0.00774 \dots

f^{^{'}} (u_{3}) = 2 \cdot 3.72836 \dots - 9.56047 \dots = - 2.10374 \dots

u_{4} = 3.73204 \dots

Δ u_{4} = ∣ 3.73204 \dots - 3.72836 \dots ∣ = 0.00367 \dots

Δ u_{4} = 0.00367 \dots

u_{5} = 3.73205 \dots : Δ u_{5} = 6.45822 E - 6

f (u_{4}) = 3.73204 \dots^{2} - 9.56047 \dots \cdot 3.73204 \dots + 21.75198 \dots = 0.00001 \dots

f^{^{'}} (u_{4}) = 2 \cdot 3.73204 \dots - 9.56047 \dots = - 2.09638 \dots

u_{5} = 3.73205 \dots

Δ u_{5} = ∣ 3.73205 \dots - 3.73204 \dots ∣ = 6.45822 E - 6

Δ u_{5} = 6.45822 E - 6

u_{6} = 3.73205 \dots : Δ u_{6} = 1.98957 E - 11

f (u_{5}) = 3.73205 \dots^{2} - 9.56047 \dots \cdot 3.73205 \dots + 21.75198 \dots = 4.17089 E - 11

f^{^{'}} (u_{5}) = 2 \cdot 3.73205 \dots - 9.56047 \dots = - 2.09637 \dots

u_{6} = 3.73205 \dots

Δ u_{6} = ∣ 3.73205 \dots - 3.73205 \dots ∣ = 1.98957 E - 11

Δ u_{6} = 1.98957 E - 11

u \approx 3.73205 \dots

Apply long division:

\frac{u ^{2} - 9.56047 \dots u + 21.75198 \dots}{u - 3.73205 \dots} = u - 5.82842 \dots

u - 5.82842 \dots \approx 0

u \approx 5.82842 \dots

The solutions are

u \approx 0.17157 \dots, u \approx 0.26794 \dots, u \approx 3.73205 \dots, u \approx 5.82842 \dots

u \approx 0.17157 \dots, u \approx 0.26794 \dots, u \approx 3.73205 \dots, u \approx 5.82842 \dots

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

(\frac{u - u ^{- 1}}{2})^{2} - 5 \frac{u + u ^{- 1}}{2}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u \approx 0.17157 \dots, u \approx 0.26794 \dots, u \approx 3.73205 \dots, u \approx 5.82842 \dots

u \approx 0.17157 \dots, u \approx 0.26794 \dots, u \approx 3.73205 \dots, u \approx 5.82842 \dots

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = 0.17157 \dots : x = ln (0.17157 \dots)

e^{x} = 0.17157 \dots

Apply exponent rules

e^{x} = 0.17157 \dots

f (x) = g (x)

, then

ln (f (x)) = ln (g (x))

ln (e^{x}) = ln (0.17157 \dots)

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (0.17157 \dots)

x = ln (0.17157 \dots)

Solve

e^{x} = 0.26794 \dots : x = ln (0.26794 \dots)

e^{x} = 0.26794 \dots

Apply exponent rules

e^{x} = 0.26794 \dots

f (x) = g (x)

, then

ln (f (x)) = ln (g (x))

ln (e^{x}) = ln (0.26794 \dots)

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (0.26794 \dots)

x = ln (0.26794 \dots)

Solve

e^{x} = 3.73205 \dots : x = ln (3.73205 \dots)

e^{x} = 3.73205 \dots

Apply exponent rules

e^{x} = 3.73205 \dots

f (x) = g (x)

, then

ln (f (x)) = ln (g (x))

ln (e^{x}) = ln (3.73205 \dots)

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (3.73205 \dots)

x = ln (3.73205 \dots)

Solve

e^{x} = 5.82842 \dots : x = ln (5.82842 \dots)

e^{x} = 5.82842 \dots

Apply exponent rules

e^{x} = 5.82842 \dots

f (x) = g (x)

, then

ln (f (x)) = ln (g (x))

ln (e^{x}) = ln (5.82842 \dots)

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (5.82842 \dots)

x = ln (5.82842 \dots)

x = ln (0.17157 \dots), x = ln (0.26794 \dots), x = ln (3.73205 \dots), x = ln (5.82842 \dots)

x = ln (0.17157 \dots), x = ln (0.26794 \dots), x = ln (3.73205 \dots), x = ln (5.82842 \dots)

Popular Examples

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